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Parikh vectors
8OYH_1 6IDS_1 4QKJ_1 Letter Amino acid
24 10 7 R Arginine
8 4 6 C Cysteine
22 9 7 E Glutamic acid
19 39 7 I Isoleucine
14 8 8 K Lycine
45 55 8 A Alanine
47 39 11 G Glycine
38 25 7 T Threonine
17 13 4 Y Tyrosine
29 9 4 N Asparagine
9 24 7 F Phenylalanine
35 26 12 S Serine
11 11 7 W Tryptophan
31 5 8 D Aspartic acid
18 22 9 Q Glutamine
19 5 10 H Histidine
36 70 7 L Leucine
6 20 0 M Methionine
21 16 4 P Proline
31 31 2 V Valine 

8OYH_1|Chain A|Furin|Homo sapiens (9606)
>6IDS_1|Chain A|MATE family efflux transporter|Vibrio cholerae (666)
>4QKJ_1|Chain A|C-type lectin domain family 2 member D|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8OYH , Knot 198 480 0.85 40 245 450 
DVYQEPTDPKFPQQWYLSGVTQRDLNVKAAWAQGYTGHGIVVSILDDGIEKNHPDLAGNYDPGASFDVNDQDPDPQPRYTQMNDNRHGTRCAGEVAAVANNGVCGVGVAYNARIGGVRMLDGEVTDAVEARSLGLNPNHIHIYSASWGPEDDGKTVDGPARLAEEAFFRGVSQGRGGLGSIFVWASGNGGREHDSCNCDGYTNSIYTLSISSATQFGNVPWYSEACSSTLATTYSSGNQNEKQIVTTDLRQKCTESHTGTSASAPLAAGIIALTLEANKNLTWRDMQHLVVQTSKPAHLNANDWATNGVGRKVSHSYGYGLLDAGAMVALAQNWTTVAPQRKCIIDILTEPKDIGKRLEVRKTVTACLGEPNHITRLEHAQARLTLSYNRRGDLAIHLVSPMGTRSTLLAARPHDYSADGFNDWAFMTTHSWDEDPSGEWVLEIENTSEANNYGTLTKFTLVLYGTASGSLVPRGSHHHH
6IDS , Knot 173 441 0.79 40 198 396 
MAMQTSTSSLAKQLFQMTWPMLFGVLSLMSFQLVNSAFIGQLGVLPLAAQGFTMPIQMVIIGIQVGLGIATTAVISRAIGAGKTEYAKQLGGLVIVIGGIGVALIALVLYLLRQPLLGLLGAPETVFAIIDHYWLWWLASAWTGAMLYFYYSVCRANGNTLLPGTLMMVTSVLNLILDPIFIFTFDLGIDGAAIATIIAFGVGIAIVAPKVAQRQWTSYQWQDLNISQSLTALGHIMGPAMLSQLLPPLSSMFATKLLASFGTAAVAAWALGSRFEFFALVAVLAMTMSLPPMIGRMLGAKEITHIRQLVRIACQFVLGFQLLIALVTYVFATPLAELMTSETEVSQILNLHLVIVPISLGALGICMLMVSVANALGKSYVALTISALRLFAFYLPCLWLGAHFYGIEGLFIGALVGNIIAGWAAWLAYQKALRSENLYFQ
4QKJ , Knot 65 135 0.78 38 107 126 
ITGQAACPESWIGFQRKCFYFSDDTKNWTSSQRFCDSQDADLAQVESFQELNFLLRYKGPSDHWIGLSREQGQPWKWINGTEWTRQFPILGAGECAYLNDKGASSARCYTERKWICSKSDIHVGTKHHHHHHHHG

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8OYH_1)}(2) \setminus P_{f(6IDS_1)}(2)|=112\), \(|P_{f(6IDS_1)}(2) \setminus P_{f(8OYH_1)}(2)|=65\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:010001001011001010110000101011110100101111011001100001011100011101010000101010000100000100011011111001101111100101111011010100110100111010010100101110001001011101100111011001011110111110101100000000010000100101001001101110001000011000001000000110001000000000100101111111111010100010100100111000011010100110011100100001011101111111100100111000011011001001100101000101011010010010010101010000010111011011100001111010000101100111100001000101011101000001000101001011101010101110100000
Pair \(Z_2\) Length of longest common subsequence
8OYH_1,6IDS_1 177 4
8OYH_1,4QKJ_1 206 4
6IDS_1,4QKJ_1 201 3

Newick tree

 
[
	4QKJ_1:10.80,
	[
		8OYH_1:88.5,6IDS_1:88.5
	]:17.30
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{921 }{\log_{20} 921}-\frac{441}{\log_{20}441})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 8OYH_1 6IDS_1 163 153.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]